Excellent Answer for a Gravitational Question On Explosion --- #No need of Energy Conservation #2 step solution using Mechanics (Pathfinder Solution)

An asteroid of mass M explodes into a spherical homogeneous cloud in free space. Due to energy received by explosion, the cloud expands and the expansion is spherically symmetric. At an instant, when radius of the cloud is Ro, all of its particles on the surface are observed receding radially away from the center of the cloud with velocity Vo. What will the radius of the cloud be, when its expansion ceases.  

The asteroid explodes into spherical homogeneous cloud and at some particular time consider the thin spherical shell around the exploded cloud at that time.

Consider the small spherical shell of mass dM ~ 0 outside the mass M. The only force acting on the shell is due to the mass of cloud M inside. 

So, 

F = GM dM / r²

Also, F on small mass element (shell) =  mass of shell * v dv / dr 

(dM) v dv/dr = GM dM / r²

Cancelling dM on both side

We get 

v dv = GM dr/ r²

 Integrate on both side; v from Vo to zero and R from Ro to R final

Solve and find the answer:

Ans is R final = 2 GMRo / (2GM - Vo²Ro)